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Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

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Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#61

Post by jacobolus »

sannee wrote: post #59
Myrmornis wrote: post #49

Earlier quoted context omitted.

Certainly integration of trigonometric functions, requiring trigonometric identities, seem to get more time allotted to them than the pedagogical value they provide AFAICS, but I'd be interested to hear someone pointing out what I'm missing there.

Trigonometry was important (presumably much more than calculus) for navigation/military applications at some point in history, so maybe it just stuck that way?

Trigonometry† is still super important for all of physics and a great deal of higher mathematics.

The problem with it is that before the past 150–200 years, people didn’t have an adequate collectionn of mathematical concepts / tools / number-like objects to work with, so the description is done in an extraordinarily unnatural and cumbersome form.

Additionally, the modern lessons are almost entirely anachronistic: the reason that people cared about trigonometric identities was that they did all calculation by hand or using pre-computed lookup tables. In that context, judicious application of some identities could save hundreds of hours of labor by a semi-skilled human computer, by reducing the number of arithmetic calculations and/or table lookups. In an age of computers this is not really a consideration, and modern students don’t have any appreciation for the purpose or context of the tools of classical trigonometry.

The classical trigonometry course should have been ripped out and fixed 50 years ago if not before (replaced by courses in vectors, complex arithmetic, and the complex exponential and logarithm). Such deeply entrenched school curricula are very difficult to modify though.

†: Note, by trigonometry what we really mean is «the relations between uniform circular motion and/or lengths of circular arcs and a square grid coordinate system»; trigonometry is something of a misnomer as there is only partial overlap between metrical 3-gon (a.k.a. “triangle”) geometry and the study of uniform circular motion, and most of the interesting parts of metrical 3-gon geometry are not covered in a trigonometry course.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#62

Post by dotancohen »

uoaei wrote: post #5
stephengillie wrote: post #2

Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.

Interestingly, children as young as 5 show an aptitude for understanding overarching concepts of calculus.[1] This makes sense: it is much easier to talk about "rates of change" and "accumulation" in simple terms and show how they are related using models that appeal to children. We don't need to dive right in to the notation and algebraic manipulations to get across the basic idea. That can come later when children…

My oldest, while in second grade, learned enough calculus to determine the _location_ of a train given it's acceleration and since it started. That's because she was interested, and asking questions, and luckily I had explained to her the slope of a graph just a few days before the train ride. She didn't learn the formula to memorize, but rather the concepts. Only then did we do the calculations the long way, on paper, pen in _her_ hand.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#63

Post by archgoon »

stephengillie wrote: post #37
CalChris wrote: post #32

Earlier quoted context omitted.

Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later. One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick ( EE 120 Linear Systems ) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to l…

I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube. We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?

>We can use complex numbers to describe a 2-dimensional number-space with a single digit.

No we can't. What digit represents 1 + 1j?

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#64

Post by rocqua »

conistonwater wrote: post #10
rocqua wrote: post #9

Earlier quoted context omitted.

I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.

Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a…

Integration can definitely not be applied to basically anything. See the entire subject of measure theory, the concepts of Riemann, Riemann-stieltjes and Lebesgue integrals.

The class of C(1) functions is quite easy. The class of intergrable functions is much more difficult. All we know is that it is larger. Consider this: to prove a function isn't differentiable, you need only give a single point where the derivative as a limit doesn't converge. To prove a function has no integral, you need to consider all possible partitions of that function's domain. (You also need to specify what exact measure is being used, etc).

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#65

Post by [deleted] »

jessaustin wrote: post #15
adamnemecek wrote: post #12

Earlier quoted context omitted.

Geometry proofs aren't real proofs.

By that measure, most proofs seen in early undergraduate years are also not real (e.g. the proofs given for the fundamental theorem of calculus). The impression one gets from GP is that they needed to see some proofs in high school. If they had taken geometry with me in Mr. Schardt's class they would have gotten their fill...

[deleted]

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#66

Post by jacobolus »

analog31 wrote: post #27
stephengillie wrote: post #2

Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.

While we're talking about turning school math on its head... I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products. Learning about vectors in physics finally got me comfortable with trig. Now, one thing we did learn in trig was to get much more proficient with algebraic manipulation. Amusingly, with complex numbers, proving trig identities becomes a trivial algebra pr…

To elaborate, “angle measure” (i.e. circular arclength corresponding to a particular rotation) is a derived quantity, not the primary abstraction we should be thinking about.

The way I think about it is that angle measure is the logarithm of a rotation, with the information about the orientation of the plane of rotation stripped out. Composition of rotations is an inherently multiplicative kind of structure, and for something a computer can understand the best representation is usually a unit-magnitude complex number. We can treat it additively by taking the logarithm, in precisely the same way we can treat scaling additively by taking the logarithm.

Symbolically, iθ = log(z), where z = x + yi is a complex number with x^2 + y^2 = 1.

This tool is very useful if you want to e.g. smoothly interpolate between rotations, but often dramatic overkill. For many problems it’s better to deal with rotations in pure vector terms, and never bother with angle measure whatsoever.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#67

Post by jacobolus »

archgoon wrote: post #63
stephengillie wrote: post #37

Earlier quoted context omitted.

I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube. We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?

>We can use complex numbers to describe a 2-dimensional number-space with a single digit. No we can't. What digit represents 1 + 1j?

There are many ways to do this: https://en.wikipedia.org/wiki/Complex-base_system

But it seems clear that the grandparent poster was using the wrong word (“digit”) and likely meant something like “we can conceive of a 2-dimensional quantity as a single number-like entity”.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#68

Post by falcor84 »

I'm not sure how well it would fit as a first course, but the thing that really helped me understand what's behind differential equations is Steven Strogatz's wonderful book "Nonlinear Dynamics And Chaos"[0].

[0] https://www.amazon.com/gp/aw/d/0813349109/

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#69

Post by fmap »

And 20 years later, this essay is still as relevant as the day it was written. I agree with pretty much everything that's in the essay, except for a few small points.

> There is nothing wrong with keeping the functional notation for density functions – as physicists and engineers always did – as long as one bears in mind that density functions cannot be evaluated, but only integrated.

This always bothered me, since, as noted in the very next section, distributions don't have an analogue to pointwise multiplication. Even worse, there is a perfectly servicable notation for such "dual functions/vectors" that physicists have been using throughout the second half of the 20th century. We could just use a consistent notation and not confuse new students, but no. "It's always been done this way" is a terrible argument and leads us to the confusing mess of notations that people still use for integrals and integral transforms...

---

Apart from that I would teach people recurrence equations/stream calculus before going into the limiting case of differential equations. It's true that differential equations are sometimes easier to handle analytically, but this is neither relevant (as the article notes) nor a great point in their favor, since we just end up teaching students a bag of tricks instead of explaining why something works...

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#70

Post by jkabrg »

mtzet wrote: post #51
stephengillie wrote: post #37

Earlier quoted context omitted.

I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube. We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?

> We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space? The complex numbers equips the vector space of 2-dimensional real numbers with a multiplication. This structure is known as an algebra. The analogue for 4 dimensions is the Quaternion Algebra, which is no longer comm…

[technical stuff] An algebra is just a vector space V with a bilinear operation * :VxV->V. [/technical stuff] There are plenty of interesting and intuitive algebras other than the stuff you get from the Cayley-Dickson constructions.

Some interesting algebras:

* you can have some fun making the imaginary unit i square to 0 instead of -1. The resulting algebra is called the dual numbers, and has some surprising properties. You can gain a bit of understanding of it by using analogies with the complex numbers.

* You can make i square to +1 without being equal to +1 or -1. Again you can gain some understanding using analogies with the complex numbers. It's perhaps not the most useful examples because it's isomorphic to R \oplus R.

* Another family of examples, a very important one actually, is the algebra of 2x2 or 3x3 or nxn matrices over the real numbers or the complex numbers.

The first two examples are useful for understanding the general concept of a quotient ring. They're hardly exhaustive, but they are easy to picture.

Now I can return to your claim about CD: What makes the Cayley-Dickson family significant is that it produces all the division algebras over the real numbers, which are the algebras for which division by nonzero elements is always possible.

[edit]

Needed to escape an asterisk. Third time lucky.

Corrected grammar.

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